This problem challenges you to find out how many odd numbers there are between pairs of numbers. Can you find a pair of numbers that has four odds between them?

In this game for two players, the idea is to take it in turns to choose 1, 3, 5 or 7. The winner is the first to make the total 37.

If there are 3 squares in the ring, can you place three different numbers in them so that their differences are odd? Try with different numbers of squares around the ring. What do you notice?

Watch the video of Fran re-ordering these number cards. What do you notice? Try it for yourself. What happens?

Ben and his mum are planting garlic. Use the interactivity to help you find out how many cloves of garlic they might have had.

In this problem we are looking at sets of parallel sticks that cross each other. What is the least number of crossings you can make? And the greatest?

Can you work out how to win this game of Nim? Does it matter if you go first or second?

Are these statements relating to odd and even numbers always true, sometimes true or never true?

Frances and Rishi were given a bag of lollies. They shared them out evenly and had one left over. How many lollies could there have been in the bag?

Find a route from the outside to the inside of this square, stepping on as many tiles as possible.

Use your addition and subtraction skills, combined with some strategic thinking, to beat your partner at this game.

Sweets are given out to party-goers in a particular way. Investigate the total number of sweets received by people sitting in different positions.

Strike it Out game for an adult and child. Can you stop your partner from being able to go?

Stop the Clock game for an adult and child. How can you make sure you always win this game?

Take a look at the video of this trick. Can you perform it yourself? Why is this maths and not magic?

In this calculation, the box represents a missing digit. What could the digit be? What would the solution be in each case?

This task encourages you to investigate the number of edging pieces and panes in different sized windows.

Florence, Ethan and Alma have each added together two 'next-door' numbers. What is the same about their answers?

This is a game for two players. Can you find out how to be the first to get to 12 o'clock?

Got It game for an adult and child. How can you play so that you know you will always win?

Watch this film carefully. Can you find a general rule for explaining when the dot will be this same distance from the horizontal axis?

This challenge asks you to imagine a snake coiling on itself.

Find out what a "fault-free" rectangle is and try to make some of your own.

This challenge focuses on finding the sum and difference of pairs of two-digit numbers.

This challenge is about finding the difference between numbers which have the same tens digit.

This challenge, written for the Young Mathematicians' Award, invites you to explore 'centred squares'.

Here are two kinds of spirals for you to explore. What do you notice?

Investigate the sum of the numbers on the top and bottom faces of a line of three dice. What do you notice?

Only one side of a two-slice toaster is working. What is the quickest way to toast both sides of three slices of bread?

Watch this video to see how to roll the dice. Now it's your turn! What do you notice about the dice numbers you have recorded?

Does this 'trick' for calculating multiples of 11 always work? Why or why not?

Two children made up a game as they walked along the garden paths. Can you find out their scores? Can you find some paths of your own?

How many ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...?

Can you see why 2 by 2 could be 5? Can you predict what 2 by 10 will be?

Nim-7 game for an adult and child. Who will be the one to take the last counter?

Use the interactivity to investigate what kinds of triangles can be drawn on peg boards with different numbers of pegs.

Find the sum and difference between a pair of two-digit numbers. Now find the sum and difference between the sum and difference! What happens?

Think of a number, square it and subtract your starting number. Is the number you’re left with odd or even? How do the images help to explain this?

A game for two people, or play online. Given a target number, say 23, and a range of numbers to choose from, say 1-4, players take it in turns to add to the running total to hit their target.

This task follows on from Build it Up and takes the ideas into three dimensions!

Ben’s class were cutting up number tracks. First they cut them into twos and added up the numbers on each piece. What patterns could they see?

Four bags contain a large number of 1s, 3s, 5s and 7s. Pick any ten numbers from the bags above so that their total is 37.

Are these statements always true, sometimes true or never true?

Can you dissect an equilateral triangle into 6 smaller ones? What number of smaller equilateral triangles is it NOT possible to dissect a larger equilateral triangle into?

Can you find all the ways to get 15 at the top of this triangle of numbers? Many opportunities to work in different ways.

This challenge encourages you to explore dividing a three-digit number by a single-digit number.

The NRICH team are always looking for new ways to engage teachers and pupils in problem solving. Here we explain the thinking behind maths trails.

Take a counter and surround it by a ring of other counters that MUST touch two others. How many are needed?

Watch this animation. What do you see? Can you explain why this happens?

This activity involves rounding four-digit numbers to the nearest thousand.